Physics I For Dummies. Steven Holzner

(

, the Greek letter delta, means “change in”), is equal to the final x position minus the initial x position. If the golf ball starts at the center of the graph — the origin of the graph, location (0, 0) — you have a change in the x location of

      The change in the y location is

If you’re more interested in figuring out the magnitude (size) of the displacement than in the changes in the x and y locations of the golf ball, that’s a different story. The question now becomes: How far is the golf ball from its starting point at the center of the graph?

      

Using the distance formula — which is just the Pythagorean theorem solved for the hypotenuse — you can find the magnitude of the displacement of the golf ball, which is the distance it travels from start to finish. The Pythagorean theorem states that the sum of the squares of the legs of a right triangle

is equal to the square on the hypotenuse

. Here, the legs of the triangle are

and

, and the hypotenuse is s. Here’s how to work the equation:

      So in this case, the magnitude of the ball’s displacement is exactly 5 meters.

      Determining direction

      

You can find the direction of an object’s movement from the values of

and

. Because these are just the legs of a right triangle, you can use basic trigonometry to find the angle of the ball’s displacement from the x-axis. The tangent of this angle is simply given by

      Therefore, the angle itself is just the inverse tangent of that:

      The ball (refer to Figure 3-2) has moved at an angle of 37° from the x-axis.

Speed Specifics: What Is Speed, Anyway?

      There’s more to the story of motion than just the actual movement. When displacement takes place, it happens in a certain amount of time. You may already know that speed is distance traveled per a certain amount of time:

      For example, if you travel distance s in a time t, your speed, v, is

      

The variable v really stands for velocity, but true velocity also has a direction associated with it, whereas speed does not. For that reason, velocity is a vector (you usually see the velocity vector represented as v or

. Vectors have both a magnitude (size) and a direction, so with velocity, you know not only how fast you’re going but also in what direction. Speed is only a magnitude (if you have a certain velocity vector, in fact, the speed is the magnitude of that vector), so you see it represented by the term v (not in bold). You can read more about velocity and displacement as vectors in Chapter 4.

      

Just as you can measure displacement, you can measure the difference in time from the beginning to the end of the motion, and you usually see it written like this:

. Technically speaking (physicists love to speak technically), velocity is the change in position (displacement) divided by the change in time, so you can also represent it like this, if, say, you’re moving along the x-axis:

      Speed can take many forms, which you find out about in the following sections.

      Reading the speedometer: Instantaneous speed

      You already have an idea of what speed is; it’s what you measure on your car’s speedometer, right? When you’re tooling along, all you have to do to see your speed is look down at the speedometer. There you have it: 75 miles per hour. Hmm, better slow it down a little — 65 miles per hour now. You’re looking at your speed at this particular moment. In other words, you see your instantaneous speed.

Instantaneous speed is an important term in understanding the physics of speed, so keep it in mind. If you’re going 65 mph right now, that’s your instantaneous speed. If you accelerate to 75 mph, that becomes your instantaneous speed. Instantaneous speed is your speed at a particular instant of time. Two seconds from now, your instantaneous speed may be totally different.

      Staying steady: Uniform speed

      What if you keep driving 65 miles per hour forever? You achieve uniform speed in physics (also called constant speed). Uniform motion is the simplest speed variation to describe, because it never changes.

      Uniform speed may be possible in the western portion of the United States, where the roads stay in straight lines for a long time and you don’t have to change your speed. But uniform speed is also possible when you drive around a circle, too. Imagine driving around a racetrack; your velocity would change (because of the constantly changing direction), but your speed could remain constant as long as you keep your gas pedal pressed down the same amount. We discuss uniform circular motion in Chapter 7, but in this chapter, we stick to motion in straight lines.

      Shifting speeds: Nonuniform motion

      Nonuniform motion varies over time; it’s the kind of speed you encounter more often in the real world. When you’re driving, for example, you change speed often, and your changes in speed come to life in an equation like this, where v_f is your final speed and v_i is your initial speed:

      The last part of this chapter is all about acceleration, which occurs in nonuniform motion. There, you see how changing speed is related to acceleration — and how you can accelerate even